Energy balance in a model of the dynamics two-dimensional baroclinic atmosphere

Integral functionals are introduced to construct functions of time that represent the upper and lower bounds for the potential energy in the system of equations of the dynamics of a two-dimensional baroclinic atmosphere. Smooth functions that are constant outside of a circle of a finite radius are considered as solutions of this system. These estimates lead to the conclusion that the unavailable potential energy and, in some cases, unavailable kinetic energy are present. The estimates are most accurate for the equations of motion of an incompressible but horizontally inhomogeneous (in density) fluid. In this case, a vector field is constructed explicitly that is orthogonal, in a sense, to the solution at any time.